Average Error: 17.1 → 0.4
Time: 45.4s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \left(\ell \cdot \left(2 + \ell \cdot \left(\frac{1}{3} \cdot \ell\right)\right) + {\ell}^{5} \cdot \frac{1}{60}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\]
double f(double J, double l, double K, double U) {
        double r11281244 = J;
        double r11281245 = l;
        double r11281246 = exp(r11281245);
        double r11281247 = -r11281245;
        double r11281248 = exp(r11281247);
        double r11281249 = r11281246 - r11281248;
        double r11281250 = r11281244 * r11281249;
        double r11281251 = K;
        double r11281252 = 2.0;
        double r11281253 = r11281251 / r11281252;
        double r11281254 = cos(r11281253);
        double r11281255 = r11281250 * r11281254;
        double r11281256 = U;
        double r11281257 = r11281255 + r11281256;
        return r11281257;
}
↓
double f(double J, double l, double K, double U) {
        double r11281258 = U;
        double r11281259 = l;
        double r11281260 = 2.0;
        double r11281261 = 0.3333333333333333;
        double r11281262 = r11281261 * r11281259;
        double r11281263 = r11281259 * r11281262;
        double r11281264 = r11281260 + r11281263;
        double r11281265 = r11281259 * r11281264;
        double r11281266 = 5.0;
        double r11281267 = pow(r11281259, r11281266);
        double r11281268 = 0.016666666666666666;
        double r11281269 = r11281267 * r11281268;
        double r11281270 = r11281265 + r11281269;
        double r11281271 = K;
        double r11281272 = r11281271 / r11281260;
        double r11281273 = cos(r11281272);
        double r11281274 = J;
        double r11281275 = r11281273 * r11281274;
        double r11281276 = r11281270 * r11281275;
        double r11281277 = r11281258 + r11281276;
        return r11281277;
}


\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
U + \left(\ell \cdot \left(2 + \ell \cdot \left(\frac{1}{3} \cdot \ell\right)\right) + {\ell}^{5} \cdot \frac{1}{60}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.1

    \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell + \left(\frac{1}{3} \cdot {\ell}^{3} + \frac{1}{60} \cdot {\ell}^{5}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  3. Simplified0.4

    \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right) + 2\right) + {\ell}^{5} \cdot \frac{1}{60}\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
  4. Using strategy rm

  5. Applied add-log-exp0.6

    \[\leadsto \left(J \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right) + 2\right) + \color{blue}{\log \left(e^{{\ell}^{5} \cdot \frac{1}{60}}\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
  6. Using strategy rm

  7. Applied pow10.6

    \[\leadsto \left(J \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right) + 2\right) + \log \left(e^{{\ell}^{5} \cdot \frac{1}{60}}\right)\right)\right) \cdot \color{blue}{{\left(\cos \left(\frac{K}{2}\right)\right)}^{1}} + U\]

  8. Applied pow10.6

    \[\leadsto \color{blue}{{\left(J \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right) + 2\right) + \log \left(e^{{\ell}^{5} \cdot \frac{1}{60}}\right)\right)\right)}^{1}} \cdot {\left(\cos \left(\frac{K}{2}\right)\right)}^{1} + U\]

  9. Applied pow-prod-down0.6

    \[\leadsto \color{blue}{{\left(\left(J \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right) + 2\right) + \log \left(e^{{\ell}^{5} \cdot \frac{1}{60}}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right)}^{1}} + U\]

  10. Simplified0.4

    \[\leadsto {\color{blue}{\left(\left(\frac{1}{60} \cdot {\ell}^{5} + \left(2 + \left(\frac{1}{3} \cdot \ell\right) \cdot \ell\right) \cdot \ell\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)}}^{1} + U\]

  11. Final simplification0.4

    \[\leadsto U + \left(\ell \cdot \left(2 + \ell \cdot \left(\frac{1}{3} \cdot \ell\right)\right) + {\ell}^{5} \cdot \frac{1}{60}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\]

Reproduce

herbie shell --seed 2019101 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))